A253974 Number of 2n-move closed giraffe paths on an unbounded chessboard from a given square to the same square.

1, 8, 168, 5120, 190120, 7964208, 362370624, 17532536736, 889716433320, 46887220540160, 2546408317827088, 141659449976239104, 8033749056463329472, 462687411167492828000, 26980019699392099317600, 1589091557661690119997120, 94361786346423775855372200

Offset: 0

Author

Vaclav Kotesovec, Jan 31 2015

Keywords

• nonn

Comments

Giraffe is a (fairy chess) leaper [1,4].

Conjecture: Number of 2n-move closed paths of leaper [r,s] on an unbounded chessboard, where 0 < r < s and gcd(r,s)=1, is asymptotic to 2^(6*n+1) / ((r^2+s^2)*Pi*n) if r+s is even, and 2^(6*n) / ((r^2+s^2)*Pi*n) if r+s is odd.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..552
• Vaclav Kotesovec, Conjectured recurrence (of order 8)
• Vaclav Kotesovec, Examples of closed giraffe paths
• Wikipedia, Fairy chess piece

Crossrefs

Cf. A094061, A254129, A254459, A307347.

Programs

Maple

b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>5*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 1],
      [1, 4], [-4, 1], [-1, 4], [4, -1], [1, -4], [-4, -1], [-1, -4]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25); # after Alois P. Heinz
# second Maple program:
poly:=expand((x*y^4+x^4*y+y^4/x+y/x^4+x/y^4+x^4/y+1/(x*y^4)+1/(x^4*y))^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do: # Vaclav Kotesovec, Apr 03 2019

Mathematica

b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 4n || x + y > 5n, 0, If[n == 0, 1, Sum[b[n - 1, Abs[x + l[[1]]], Abs[y + l[[2]]]], {l, {{4, 1}, {1, 4}, {-4, 1}, {-1, 4}, {4, -1}, {1, -4}, {-4, -1}, {-1, -4}}}]]];
a[n_] := b[2n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 01 2020, after Maple *)

Formula

a(n) ~ 64^n / (17*Pi*n).

a(n) = the constant term in the expansion of (x*y^4 + x^4*y + 1/x*y^4 + 1/x^4*y + x/y^4 + x^4/y + 1/x/y^4 + 1/x^4/y)^(2*n). - Vaclav Kotesovec, Apr 01 2019